3.2524 \(\int \frac{5-x}{(3+2 x)^4 (2+5 x+3 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=174 \[ -\frac{2 (47 x+37)}{5 (2 x+3)^3 \left (3 x^2+5 x+2\right )^{3/2}}+\frac{9696 \sqrt{3 x^2+5 x+2}}{625 (2 x+3)}+\frac{1048 \sqrt{3 x^2+5 x+2}}{15 (2 x+3)^2}+\frac{47552 \sqrt{3 x^2+5 x+2}}{375 (2 x+3)^3}+\frac{12 (638 x+603)}{25 (2 x+3)^3 \sqrt{3 x^2+5 x+2}}+\frac{46108 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{625 \sqrt{5}} \]

[Out]

(-2*(37 + 47*x))/(5*(3 + 2*x)^3*(2 + 5*x + 3*x^2)^(3/2)) + (12*(603 + 638*x))/(25*(3 + 2*x)^3*Sqrt[2 + 5*x + 3
*x^2]) + (47552*Sqrt[2 + 5*x + 3*x^2])/(375*(3 + 2*x)^3) + (1048*Sqrt[2 + 5*x + 3*x^2])/(15*(3 + 2*x)^2) + (96
96*Sqrt[2 + 5*x + 3*x^2])/(625*(3 + 2*x)) + (46108*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(625*
Sqrt[5])

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Rubi [A]  time = 0.115879, antiderivative size = 174, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {822, 834, 806, 724, 206} \[ -\frac{2 (47 x+37)}{5 (2 x+3)^3 \left (3 x^2+5 x+2\right )^{3/2}}+\frac{9696 \sqrt{3 x^2+5 x+2}}{625 (2 x+3)}+\frac{1048 \sqrt{3 x^2+5 x+2}}{15 (2 x+3)^2}+\frac{47552 \sqrt{3 x^2+5 x+2}}{375 (2 x+3)^3}+\frac{12 (638 x+603)}{25 (2 x+3)^3 \sqrt{3 x^2+5 x+2}}+\frac{46108 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{625 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]

Int[(5 - x)/((3 + 2*x)^4*(2 + 5*x + 3*x^2)^(5/2)),x]

[Out]

(-2*(37 + 47*x))/(5*(3 + 2*x)^3*(2 + 5*x + 3*x^2)^(3/2)) + (12*(603 + 638*x))/(25*(3 + 2*x)^3*Sqrt[2 + 5*x + 3
*x^2]) + (47552*Sqrt[2 + 5*x + 3*x^2])/(375*(3 + 2*x)^3) + (1048*Sqrt[2 + 5*x + 3*x^2])/(15*(3 + 2*x)^2) + (96
96*Sqrt[2 + 5*x + 3*x^2])/(625*(3 + 2*x)) + (46108*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(625*
Sqrt[5])

Rule 822

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x
)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*
c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2
*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -
b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b,
c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] ||
IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 834

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((m
+ 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[(c*d*f - f*b*e + a*e*g)*(m + 1)
+ b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] &&
NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p])

Rule 806

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2)), x] - Dist[(b
*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x],
x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[Sim
plify[m + 2*p + 3], 0]

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{5-x}{(3+2 x)^4 \left (2+5 x+3 x^2\right )^{5/2}} \, dx &=-\frac{2 (37+47 x)}{5 (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}}-\frac{2}{15} \int \frac{1473+1410 x}{(3+2 x)^4 \left (2+5 x+3 x^2\right )^{3/2}} \, dx\\ &=-\frac{2 (37+47 x)}{5 (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}}+\frac{12 (603+638 x)}{25 (3+2 x)^3 \sqrt{2+5 x+3 x^2}}+\frac{4}{75} \int \frac{33846+34452 x}{(3+2 x)^4 \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{2 (37+47 x)}{5 (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}}+\frac{12 (603+638 x)}{25 (3+2 x)^3 \sqrt{2+5 x+3 x^2}}+\frac{47552 \sqrt{2+5 x+3 x^2}}{375 (3+2 x)^3}-\frac{4 \int \frac{-222726-213984 x}{(3+2 x)^3 \sqrt{2+5 x+3 x^2}} \, dx}{1125}\\ &=-\frac{2 (37+47 x)}{5 (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}}+\frac{12 (603+638 x)}{25 (3+2 x)^3 \sqrt{2+5 x+3 x^2}}+\frac{47552 \sqrt{2+5 x+3 x^2}}{375 (3+2 x)^3}+\frac{1048 \sqrt{2+5 x+3 x^2}}{15 (3+2 x)^2}+\frac{2 \int \frac{775170+589500 x}{(3+2 x)^2 \sqrt{2+5 x+3 x^2}} \, dx}{5625}\\ &=-\frac{2 (37+47 x)}{5 (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}}+\frac{12 (603+638 x)}{25 (3+2 x)^3 \sqrt{2+5 x+3 x^2}}+\frac{47552 \sqrt{2+5 x+3 x^2}}{375 (3+2 x)^3}+\frac{1048 \sqrt{2+5 x+3 x^2}}{15 (3+2 x)^2}+\frac{9696 \sqrt{2+5 x+3 x^2}}{625 (3+2 x)}+\frac{46108}{625} \int \frac{1}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{2 (37+47 x)}{5 (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}}+\frac{12 (603+638 x)}{25 (3+2 x)^3 \sqrt{2+5 x+3 x^2}}+\frac{47552 \sqrt{2+5 x+3 x^2}}{375 (3+2 x)^3}+\frac{1048 \sqrt{2+5 x+3 x^2}}{15 (3+2 x)^2}+\frac{9696 \sqrt{2+5 x+3 x^2}}{625 (3+2 x)}-\frac{92216}{625} \operatorname{Subst}\left (\int \frac{1}{20-x^2} \, dx,x,\frac{-7-8 x}{\sqrt{2+5 x+3 x^2}}\right )\\ &=-\frac{2 (37+47 x)}{5 (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}}+\frac{12 (603+638 x)}{25 (3+2 x)^3 \sqrt{2+5 x+3 x^2}}+\frac{47552 \sqrt{2+5 x+3 x^2}}{375 (3+2 x)^3}+\frac{1048 \sqrt{2+5 x+3 x^2}}{15 (3+2 x)^2}+\frac{9696 \sqrt{2+5 x+3 x^2}}{625 (3+2 x)}+\frac{46108 \tanh ^{-1}\left (\frac{7+8 x}{2 \sqrt{5} \sqrt{2+5 x+3 x^2}}\right )}{625 \sqrt{5}}\\ \end{align*}

Mathematica [A]  time = 0.121467, size = 150, normalized size = 0.86 \[ \frac{2 \left (594400 \left (3 x^2+5 x+2\right )^2+2250 (638 x+603) \left (3 x^2+5 x+2\right )+2 (2 x+3) \left (3 x^2+5 x+2\right )^{3/2} \left (10 (7272 x+27283) \sqrt{3 x^2+5 x+2}-34581 \sqrt{5} (2 x+3)^2 \tanh ^{-1}\left (\frac{-8 x-7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )\right )-1875 (47 x+37)\right )}{9375 (2 x+3)^3 \left (3 x^2+5 x+2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(5 - x)/((3 + 2*x)^4*(2 + 5*x + 3*x^2)^(5/2)),x]

[Out]

(2*(-1875*(37 + 47*x) + 2250*(603 + 638*x)*(2 + 5*x + 3*x^2) + 594400*(2 + 5*x + 3*x^2)^2 + 2*(3 + 2*x)*(2 + 5
*x + 3*x^2)^(3/2)*(10*(27283 + 7272*x)*Sqrt[2 + 5*x + 3*x^2] - 34581*Sqrt[5]*(3 + 2*x)^2*ArcTanh[(-7 - 8*x)/(2
*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])))/(9375*(3 + 2*x)^3*(2 + 5*x + 3*x^2)^(3/2))

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Maple [A]  time = 0.011, size = 169, normalized size = 1. \begin{align*} -{\frac{151}{200} \left ( x+{\frac{3}{2}} \right ) ^{-2} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{-{\frac{3}{2}}}}-{\frac{862}{125} \left ( x+{\frac{3}{2}} \right ) ^{-1} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{-{\frac{3}{2}}}}+{\frac{11527}{750} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{-{\frac{3}{2}}}}-{\frac{11830+14196\,x}{375} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{-{\frac{3}{2}}}}+{\frac{12120+14544\,x}{625}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}}}+{\frac{23054}{625}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}}}-{\frac{46108\,\sqrt{5}}{3125}{\it Artanh} \left ({\frac{2\,\sqrt{5}}{5} \left ( -{\frac{7}{2}}-4\,x \right ){\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}}} \right ) }-{\frac{13}{120} \left ( x+{\frac{3}{2}} \right ) ^{-3} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)/(3+2*x)^4/(3*x^2+5*x+2)^(5/2),x)

[Out]

-151/200/(x+3/2)^2/(3*(x+3/2)^2-4*x-19/4)^(3/2)-862/125/(x+3/2)/(3*(x+3/2)^2-4*x-19/4)^(3/2)+11527/750/(3*(x+3
/2)^2-4*x-19/4)^(3/2)-2366/375*(5+6*x)/(3*(x+3/2)^2-4*x-19/4)^(3/2)+2424/625*(5+6*x)/(3*(x+3/2)^2-4*x-19/4)^(1
/2)+23054/625/(3*(x+3/2)^2-4*x-19/4)^(1/2)-46108/3125*5^(1/2)*arctanh(2/5*(-7/2-4*x)*5^(1/2)/(12*(x+3/2)^2-16*
x-19)^(1/2))-13/120/(x+3/2)^3/(3*(x+3/2)^2-4*x-19/4)^(3/2)

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Maxima [A]  time = 1.64724, size = 343, normalized size = 1.97 \begin{align*} -\frac{46108}{3125} \, \sqrt{5} \log \left (\frac{\sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac{5}{2 \,{\left | 2 \, x + 3 \right |}} - 2\right ) + \frac{14544 \, x}{625 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} + \frac{35174}{625 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} - \frac{4732 \, x}{125 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}} - \frac{13}{15 \,{\left (8 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x^{3} + 36 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x^{2} + 54 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x + 27 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}\right )}} - \frac{151}{50 \,{\left (4 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x^{2} + 12 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x + 9 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}\right )}} - \frac{1724}{125 \,{\left (2 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x + 3 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}\right )}} - \frac{12133}{750 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)/(3+2*x)^4/(3*x^2+5*x+2)^(5/2),x, algorithm="maxima")

[Out]

-46108/3125*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) + 14544/625*x/sqrt(
3*x^2 + 5*x + 2) + 35174/625/sqrt(3*x^2 + 5*x + 2) - 4732/125*x/(3*x^2 + 5*x + 2)^(3/2) - 13/15/(8*(3*x^2 + 5*
x + 2)^(3/2)*x^3 + 36*(3*x^2 + 5*x + 2)^(3/2)*x^2 + 54*(3*x^2 + 5*x + 2)^(3/2)*x + 27*(3*x^2 + 5*x + 2)^(3/2))
- 151/50/(4*(3*x^2 + 5*x + 2)^(3/2)*x^2 + 12*(3*x^2 + 5*x + 2)^(3/2)*x + 9*(3*x^2 + 5*x + 2)^(3/2)) - 1724/12
5/(2*(3*x^2 + 5*x + 2)^(3/2)*x + 3*(3*x^2 + 5*x + 2)^(3/2)) - 12133/750/(3*x^2 + 5*x + 2)^(3/2)

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Fricas [A]  time = 2.2739, size = 527, normalized size = 3.03 \begin{align*} \frac{2 \,{\left (34581 \, \sqrt{5}{\left (72 \, x^{7} + 564 \, x^{6} + 1862 \, x^{5} + 3355 \, x^{4} + 3560 \, x^{3} + 2223 \, x^{2} + 756 \, x + 108\right )} \log \left (\frac{4 \, \sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (8 \, x + 7\right )} + 124 \, x^{2} + 212 \, x + 89}{4 \, x^{2} + 12 \, x + 9}\right ) + 5 \,{\left (523584 \, x^{6} + 4495032 \, x^{5} + 15334836 \, x^{4} + 26717636 \, x^{3} + 25105026 \, x^{2} + 12060957 \, x + 2313929\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}}{9375 \,{\left (72 \, x^{7} + 564 \, x^{6} + 1862 \, x^{5} + 3355 \, x^{4} + 3560 \, x^{3} + 2223 \, x^{2} + 756 \, x + 108\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)/(3+2*x)^4/(3*x^2+5*x+2)^(5/2),x, algorithm="fricas")

[Out]

2/9375*(34581*sqrt(5)*(72*x^7 + 564*x^6 + 1862*x^5 + 3355*x^4 + 3560*x^3 + 2223*x^2 + 756*x + 108)*log((4*sqrt
(5)*sqrt(3*x^2 + 5*x + 2)*(8*x + 7) + 124*x^2 + 212*x + 89)/(4*x^2 + 12*x + 9)) + 5*(523584*x^6 + 4495032*x^5
+ 15334836*x^4 + 26717636*x^3 + 25105026*x^2 + 12060957*x + 2313929)*sqrt(3*x^2 + 5*x + 2))/(72*x^7 + 564*x^6
+ 1862*x^5 + 3355*x^4 + 3560*x^3 + 2223*x^2 + 756*x + 108)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{x}{144 x^{8} \sqrt{3 x^{2} + 5 x + 2} + 1344 x^{7} \sqrt{3 x^{2} + 5 x + 2} + 5416 x^{6} \sqrt{3 x^{2} + 5 x + 2} + 12296 x^{5} \sqrt{3 x^{2} + 5 x + 2} + 17185 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 15126 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 8181 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 2484 x \sqrt{3 x^{2} + 5 x + 2} + 324 \sqrt{3 x^{2} + 5 x + 2}}\, dx - \int - \frac{5}{144 x^{8} \sqrt{3 x^{2} + 5 x + 2} + 1344 x^{7} \sqrt{3 x^{2} + 5 x + 2} + 5416 x^{6} \sqrt{3 x^{2} + 5 x + 2} + 12296 x^{5} \sqrt{3 x^{2} + 5 x + 2} + 17185 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 15126 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 8181 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 2484 x \sqrt{3 x^{2} + 5 x + 2} + 324 \sqrt{3 x^{2} + 5 x + 2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)/(3+2*x)**4/(3*x**2+5*x+2)**(5/2),x)

[Out]

-Integral(x/(144*x**8*sqrt(3*x**2 + 5*x + 2) + 1344*x**7*sqrt(3*x**2 + 5*x + 2) + 5416*x**6*sqrt(3*x**2 + 5*x
+ 2) + 12296*x**5*sqrt(3*x**2 + 5*x + 2) + 17185*x**4*sqrt(3*x**2 + 5*x + 2) + 15126*x**3*sqrt(3*x**2 + 5*x +
2) + 8181*x**2*sqrt(3*x**2 + 5*x + 2) + 2484*x*sqrt(3*x**2 + 5*x + 2) + 324*sqrt(3*x**2 + 5*x + 2)), x) - Inte
gral(-5/(144*x**8*sqrt(3*x**2 + 5*x + 2) + 1344*x**7*sqrt(3*x**2 + 5*x + 2) + 5416*x**6*sqrt(3*x**2 + 5*x + 2)
+ 12296*x**5*sqrt(3*x**2 + 5*x + 2) + 17185*x**4*sqrt(3*x**2 + 5*x + 2) + 15126*x**3*sqrt(3*x**2 + 5*x + 2) +
8181*x**2*sqrt(3*x**2 + 5*x + 2) + 2484*x*sqrt(3*x**2 + 5*x + 2) + 324*sqrt(3*x**2 + 5*x + 2)), x)

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Giac [A]  time = 1.19415, size = 385, normalized size = 2.21 \begin{align*} \frac{46108}{3125} \, \sqrt{5} \log \left (\frac{{\left | -4 \, \sqrt{3} x - 2 \, \sqrt{5} - 6 \, \sqrt{3} + 4 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt{3} x + 2 \, \sqrt{5} - 6 \, \sqrt{3} + 4 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}}\right ) + \frac{2 \,{\left ({\left (12 \,{\left (19992 \, x + 58207\right )} x + 636631\right )} x + 184301\right )}}{3125 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}} - \frac{8 \,{\left (296724 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{5} + 2103870 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{4} + 16891990 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 21246975 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 38063715 \, \sqrt{3} x + 8723544 \, \sqrt{3} - 38063715 \, \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}}{9375 \,{\left (2 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 6 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} + 11\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)/(3+2*x)^4/(3*x^2+5*x+2)^(5/2),x, algorithm="giac")

[Out]

46108/3125*sqrt(5)*log(abs(-4*sqrt(3)*x - 2*sqrt(5) - 6*sqrt(3) + 4*sqrt(3*x^2 + 5*x + 2))/abs(-4*sqrt(3)*x +
2*sqrt(5) - 6*sqrt(3) + 4*sqrt(3*x^2 + 5*x + 2))) + 2/3125*((12*(19992*x + 58207)*x + 636631)*x + 184301)/(3*x
^2 + 5*x + 2)^(3/2) - 8/9375*(296724*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^5 + 2103870*sqrt(3)*(sqrt(3)*x - sqrt
(3*x^2 + 5*x + 2))^4 + 16891990*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^3 + 21246975*sqrt(3)*(sqrt(3)*x - sqrt(3*x
^2 + 5*x + 2))^2 + 38063715*sqrt(3)*x + 8723544*sqrt(3) - 38063715*sqrt(3*x^2 + 5*x + 2))/(2*(sqrt(3)*x - sqrt
(3*x^2 + 5*x + 2))^2 + 6*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) + 11)^3